# Chirp Wave

Wave · y = f(x, t)

`y = sin(x² + t) · cos(3x − t)`

[Open in the app](https://www.wavelace.com/app#p=0) · [This page](https://www.wavelace.com/presets/chirp-wave)

### What it draws

The formula is a product of two waves. In `sin(x² + t)` the phase grows like `x²`, so the local wavenumber is its slope, `2x`, and crests are `π/x` apart. That is `3.14` near `x = 1` and `0.52` out at `x = 6`. That widening and narrowing is what a *chirp* is. The second factor, `cos(3x − t)`, is a plain wave with crests `2π/3 ≈ 2.09` apart, sliding right at `1/3` unit per second.

### Why the pattern

A product of a sine and a cosine is a sum of two waves: `sin(x² + t)·cos(3x − t) = ½sin(x² + 3x) + ½sin(x² − 3x + 2t)`. The first half has no `t` in it at all, so half of what is on screen is frozen. Its local wavenumber is `2x + 3`, which vanishes at `x = −1.5`. There the still half stretches into one long slow bend, while at `x = 6` its crests are only `2π/15 ≈ 0.42` apart.

The second half carries all of the motion. Its crests hold `x² − 3x + 2t` fixed, so each one slides at `2/(2x − 3)` towards `x = 1.5` from both sides. At `x = 6` that is `0.22` unit per second leftward, and at `x = −6` it is `0.13` rightward. The eye reads the two halves as one restless texture that streams inward and never repeats across the plot.

### Where chirps turn up

A signal whose frequency sweeps is the working tool of radar and sonar. A long chirp carries the energy of a long pulse and, matched against a copy of itself, the timing of a short one. Bats and dolphins sweep their calls for the same reason.

### Try

- `Front` in the deck: straight on, the crowding of the crests towards the right edge is easiest to read.
- Type the split out in full, `0.5 · sin(x² + 3x) + 0.5 · sin(x² − 3x + 2t)`: the picture does not change.
- Then keep the frozen half alone, `0.5 · sin(x² + 3x)`. Nothing moves, and the long bend at `x = −1.5` stands out.
- Change `cos(3x − t)` to `cos(6x − t)`: the still point moves to `x = −3` and the inward drift to `x = 3`.
- Drop `Speed` to 0.3 and watch one crest walk inward instead of the whole texture shimmering.

### Read more

- [Chirp](https://en.wikipedia.org/wiki/Chirp)
- [Pulse compression](https://en.wikipedia.org/wiki/Pulse_compression)
- [Instantaneous phase and frequency](https://en.wikipedia.org/wiki/Instantaneous_phase_and_frequency)
- [List of trigonometric identities](https://en.wikipedia.org/wiki/List_of_trigonometric_identities)